As we will see in the subsequent Sec. REF , only the second
choice (ii) is consistent with the definition of \(S^m_{[x]}\)  via
path probabilities, as well as with the usual IFR for the total
entropy production. It seems worthwhile noting that the introduction
of an effective, in our case time-dependent temperature is not a
new concept at all. Indeed, generalised temperatures have been used,
e.g., in weak turbulence, granular matter, and glassy material
[1]}, and more recently for active matter
[2]}, [3]}, [4]}, [5]}, [6]}, [7]}. We
remark, however, that its straightforward definition based on FDRs
has been criticised [8]}.
